2. DEFINITIONS

A Statistician: is ``a man prepared to estimate the probability that
the Sun will rise tomorrow in the light of its past performance''
(3).
This definition, which I admit to taking out of context, is by an
eminent statistician; it may confirm our worst suspicions about
statisticians eminent or otherwise. At the very least it should
emphasize care of definition - what is tomorrow when the Sun does not
rise? Let us be more careful:

``Statistics'' is a term loosely used to describe both the science and
the values. In fact, the science is

Statistical Inference: the determination of properties of the
population from the sample,

while

Statistics are values (usually, but not necessarily, numerical)
determined from some or all of the values of a sample.

Good statistics are those from which our conclusions concerning the
population are stable from sample to sample, while good samples provide
good statistics, and require appropriate design of experiment.

The ``goodness'' of the experiment, the sample, or the statistic is
indicated by the

Level of Significance: suppose we perform an experiment to distinguish
between two rival hypotheses, the null hypothesis
(H0; ``failure'', no
result, no detection, no correlation) and its alternative
(H0;
``success'', etc.). Before the experiment we make ourselves very familiar
with ``failure'' by determining, assuming H0 to be
true, the set of all
possible values of the statistic under test, the statistic we have
chosen to use in deciding between H0 and
H1. Furthermore, suppose that
when we do the experiment we obtain a value of the statistic which is
``unusual'' in comparison with this set, so unusual that, say, only 1 per
cent of all values computed under the H0 hypothesis
are so extreme. We
can then reject H0 in favour of H1
at the 1 per cent level of
significance. The level of significance is thus the probability of
rejecting H0 when it is, in fact, true.

Now consider N values of xi where i = 1,
2 . . . N and x may have a
continuous or a discrete distribution. The following definitions are
general:

1. Location measures

Median: arrange xi according to size; renumber. Then

xmed

= xj where
j = N / 2 + 0.5, N odd

= 1/2 (xj + xj+1 where
j = N / 2, N even.

Mode: xmode is the value of
xi occurring most frequently.

2. Dispersion measures

3. Moments

(Moments may be taken about any value of x; those about
the arithmetic
mean as above are termed central moments.) Note that
µ2 = 2; this and
the next few moments characterize a probability distribution (defined
below). The first two moments are useless, since
µ0 1
and µ 0.

Skewness:

1 = µ32
/ µ23 indicates deviation from symmetry;

= 0 for symmetry about µ.

Kurtosis:

2 = µ4 /
µ22 indicates degree of peakiness;

= 3 for Normal distribution.

Finally, consider probability distributions: if x is a continuous
random variable, then f (x) is its probability density
function if it meets these conditions:

(1) Probability

(2)

and

(3) f (x) is a single-valued non-negative number for all real
x. The corresponding distribution function is

Table 1.The common probability
distributions.

Distribution

Density function

Mean

Variance

Raison d'Être

Uniform

(a+b)/2

(b-a)/12

In the study of rounding errors, and as
a tool in theoretical studies of other continuous distributions.

Binomial

np

npq

x is the number of ``successes'' in an
experiment with two possible
outcomes, one (``success'') of probability p, and the other
(``failure'') of probability q = 1 - p. Becomes a Normal
distribution as n -> .

Poisson

µ

µ

The limit for the Binomial distribution as
p << 1, setting µnp. It is
the ``count-rate'' distribution, e.g. take a star from which an average
of µ photons are received per t (out of a total of n emitted; p << 1);
the probability of receiving x photons in a t is f
(x;µ). Tends to the
Normal distribution as µ -> .

Normal (Gaussian)

µ

2

The essential distribution; see text. Central
Limit Theorem ensures
that majority of ``scattered things'' are dispersed according to f
(x;µ,).

Chi-square

2

Vital in the comparison of samples, model
testing; characterizes the
dispersion of observed samples from the expected dispersion, because
if xi is a sample of variables Normally and independently distributed
with means µi and variances i, then
obeys f (2; )
Invariably tabulated and used in integral form. Tends to Normal
distribution as -> .

Student t

0

/(-2) (for > 2)

For comparison of means, Normally-distributed
populations; if n xis
are taken from a Normal population (µ,), and if xs and
s are found
as in text, then t = is
distributed as f (t,)
where ``degrees
of freedom'' = n -
1. Statistic t can also be formulated to compare
means for samples from Normal populations with same , different µ
(4). Tends to
Normal as -> .

F

2/(2-2) (for 2 > 2)

For comparison of two variances, or of more than
two means;
if two statistics (1
and 2) each follow the
Chi-square distribution,
then is distributed as
f(F;1,2). Care required in application;
see (4),
(9).

Probability densities and distribution functions may be similarly
defined for sets of discrete values x = x1,
x2. . . .xn, and for
multivariate distributions. The better-known (continuous) functions
appear in Table 1, together with location and
dispersion measures - note
that the previous definitions for these may be written in integral form
for continuous distributions. The table includes some indication of how
and/or where each distribution arises, and for most of them, I avoid
further discussion. But there is one whose rôle is so fundamental that
it cannot be treated in such a cavalier manner. This follows in the next
section.